Review of Fourth-Order Predictive Modeling and Illustrative Application to a Nuclear Reactor Benchmark. I. Typical High-Order Sensitivity and Uncertainty Analysis

Abstract

This work (in two parts) will review the recently developed predictive modeling methodology called “4th-BERRU-PM” and its applicability to nuclear energy systems as exemplified by an illustrative application to the Polyethylene-Reflected Plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The acronym 4th-BERRU-PM designates the “Fourth-Order Best-Estimate Results with Reduced Uncertainties Predictive Modeling” methodology, which uses the Maximum Entropy (MaxEnt) principle to incorporate fourth-order experimental and computational information, including fourth (and higher) order sensitivities of computed model responses to model parameters, while yielding best-estimate results with reduced uncertainties for the first fourth-order moments (mean values, covariance, skewness, and kurtosis) of the optimally predicted posterior distribution of model results and calibrated model parameters. The 4th-BERRU-PM methodology encompasses the scopes of high-order sensitivity analysis (SA), uncertainty quantification (UQ), data assimilation (DA) and model calibration (MC), as will be illustrated in this work by means of the above-mentioned OECD/NEA reactor physics benchmark. This benchmark is modeled using the neutron transport Boltzmann equation involving 21,976 imprecisely known parameters, the solution of which is representative of “large-scale computations”. The model result (“response”) of interest is the leakage of neutrons through the outer surface of this spherical benchmark, which can be computed numerically and measured experimentally. Part 1 of this work illustrates the impact of high-order sensitivities, in conjunction with parameter standard deviations of various magnitudes, on the determination of the expected value and variance of the computed response in terms of the first four moments of the distribution of the uncertain model parameters. Part 2 of this work will illustrate the capabilities of the 4th-BERRU-PM methodology for combining computational and experimental information, up to and including forth-order sensitivities and distributional moments, for producing best-estimate values for the predicted responses and model parameters while reducing their accompanying uncertainties.

1. Introduction

This work reviews the recently developed “4th-Order Best-Estimate Results with Reduced Uncertainties Predictive Modeling” (abbreviated as “4th-BERRU-PM”) methodology [1] and illustrates its application to energy systems by considering the Polyethylene-Reflected Plutonium (acronym: PERP) OECD/NEA reactor physics benchmark [2]. The 4th-BERRU-PM uses the Maximum Entropy (MaxEnt) Principle [3] to combine sensitivities and moments (up to and including fourth-order) of the distribution of model parameters and model responses (i.e., results of interest), which stem from model computations and imprecisely known external experimental measurements. Using this high-order computational and experimental information, the 4th-BERRU-PM methodology yields the 4th-order MaxEnt joint posterior distribution of model parameters and responses. In particular, this posterior distribution yields best-estimate results for the optimally predicted moments (up to and including fourth-order) of the best-estimate predicted model parameters (“model calibration”) and predicted responses. The 4th-BERRU-PM methodology encompasses the scopes of high-order sensitivity analysis (SA), uncertainty quantification (UQ), data assimilation (DA), and model calibration (MC), including, as particular cases, the results delivered by the second-order predictive modeling methodology [4], while vastly generalizing the extant data adjustment [5,6] and data assimilation [7,8,9,10,11] methodologies.

The fundamental importance of the new results provided by the 4th-BERRU-PM methodology will be highlighted in this work by applying this predictive modeling methodology to the Polyethylene-Reflected Plutonium OECD/NEA reactor physics benchmark [2]. As has been detailed in [12], the numerical modeling of the PERP benchmark is performed by using the neutron transport Boltzmann equation, involving 21,976 imprecisely known parameters. The response (i.e., result) of interest for this benchmark is the total leakage of neutrons through the benchmark’s outer surface. The computation of high-order sensitivities of the leakage response with respect to the benchmark’s parameters is representative of “large-scale computations” and has been accomplished by applying the high-order adjoint sensitivity analysis methodology detailed in [12], which overcomes the curse of dimensionality [13] in sensitivity analysis. The computational model used for determining the neutron distribution within the PERP benchmark and for determining the sensitivities (up to fourth-order) of the neutron leakage response with respect to the benchmark’s uncertain parameters has been detailed in [12] and is summarized in Appendix A for convenient referencing.

Section 2 of this work reviews the mathematical forms of the “input information” that needs to be extracted from the computational model to be incorporated into the 4th-BERRU-PM methodology. Extracting this information requires performing a “4th-order sensitivity analysis” (4th-SA) and a “4th-order uncertainty quantification” (4th-UQ) of the computational model by propagating the first four moments (including the means, variances, skewness, and kurtosis) of the distribution of model parameters using the response sensitivities (from first- to fourth-order) to obtain the requisite moments (from first to fourth) of the distribution of computed responses. This section also briefly discusses the computational issues required to alleviate the impact of the ”curse of dimensionality” [13] when computing such 4th-order sensitivities and uncertainties for energy systems.

It has been shown in Ref. [12] that for uniform relative standard deviations of 3% for the model parameters (total cross sections), the Taylor-series expansion of the computed response in terms of the variations in the model parameters is expected to be convergent, since the convergence “ratio-test” of the 3rd-order term with respect to the 2nd-order term of the Taylor series is 0.58 and the ratio of the 4th-order term with respect to the 3rd-order term of the Taylor series is 0.68. Both of these results are below 1.00; therefore, even though it is not possible to conduct a bona fide convergence test involving the ratio of the nth term and the (n + 1)th term, this Taylor series is expected to be convergent.

On the other hand, it has been shown in [12] that, for relative standard deviations of 5% for the model parameters, the ratio of the 3rd-order term with respect to the 2nd-order term of the Taylor series expansion of the computed response in terms of the model parameters is 0.97 < 1.00, but the ratio of the 4th-order term with respect to the 3rd-order term of the Taylor series is 1.13 > 1.00. These ratios indicate that relative standard deviations of 5% for the model parameters are “borderline” values for which the Taylor-series expansion of the computed response in terms of the model parameters is expected to be divergent. On the other hand, a relative standard deviation of 5% is representative of the “usual uncertainties encountered in practice” and is therefore often encountered in measurements of total cross sections. This situation, therefore, underscores the impact of the higher-order response sensitivities when dealing with parameter uncertainties that are representative of uncertainties encountered in practice while also being “borderline” in terms of the convergence of the Taylor-series expansion of the computed response in terms of the model parameters. Since this Taylor-series expansion underlies the determination of the statistics (expected values, variance, etc.) of the distribution of the computed response in the phase space of imprecisely known model parameters, its convergence or divergence must be established under all circumstances in which it is being used. Section 3 presents the application of the mathematical concepts reviewed in Section 2 to the PERP reactor physics benchmark, illustrating the effect of the high-order sensitivities on the expected value and variance of the computed model response when considering: (i) parameters that are known with “high precision” having uniform standard deviations of 2%; (ii) parameters known with “medium precision” having uniform standard deviations of 5%; and (iii) parameters known with “low precision” having uniform standard deviations of 10%, respectively. Section 4 concludes this work by discussing the impact of the combination of parameter uncertainties with high-order sensitivities on the uncertainties in the computed model and also prepares the ground for the continuation, in the accompanying Part 2 [14], of the application of the 4th-BERRU-PM methodology for obtaining the predicted best-estimate mean value, standard deviation, skewness, and kurtosis for the neutron leakage response of the PERP benchmark.

2. Model Sensitivity and Uncertainty Analysis Input for the Fourth-Order Maximum Entropy Based Predictive Modeling Methodology (4th-BERRU-PM): Review and Applicability to Energy Systems

The 4th-BERRU-PM methodology uses as “input” the first four moments of the unknown distributions of the responses computed using a mathematical/computational model, which are combined using the maximum entropy principle with the first four moments of the distribution of measured responses. The moments of the distribution of the computed model responses are obtained by combining the moments of the distribution of model parameters with the sensitivities of the model responses with respect to the model parameters. Thus, the moments of the distribution of the computed model responses are obtained by performing (simultaneously or sequentially) a 4th-order sensitivity analysis and a 4th-order uncertainty analysis of the underlying mathematical/computational model.

The mathematical expressions of the moments of the computed responses are obtained in terms of the moments of the distribution of model parameters by expanding formally each response in a Taylor-series around the nominal or mean parameter values and subsequently using this series, within its radius of convergence, for obtaining expressions of the moments of the respective responses in terms of the moments of the model parameters and the response sensitivities (i.e., derivatives) with respect to the model parameters. General expressions, up to sixth-order sensitivities, for the moments of computed responses can be found in Ref. [12], which generalizes the expressions originally obtained in Ref. [15]. In particular, the following fourth-order Taylor-series expansion of a response, denoted as $r_k\left(\alpha \right)$, as a function of the parameters $\alpha \triangleq {\left({\alpha }_1,\dots ,{\alpha }_{TP}\right)}^{†}$, where $TP$ denotes the total number of parameters under consideration, is used within the 4th-BERRU-PM methodology to compute the various moments of the distribution of the leakage response in the phase-space of the benchmark’s total cross section (parameters):

$$\begin{array}{l}{r}_k\left(\alpha \right)=r_k\left({\alpha }^0\right)+{\displaystyle \sum _{j_1=1}^{TP}{\left\{\frac{\mathsf{\partial }{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_{j_1}}\right\}}_{{\alpha }^0}\delta t_{j_1}}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j_1=1}^{TP}{\displaystyle \sum _{j_2=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^2{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_{j_1}\mathsf{\partial }{\alpha }_{j_2}}\right\}}_{{\alpha }^0}}}\delta {\alpha }_{j_1}\delta {\alpha }_{j_2}\\ +{\displaystyle \frac{1}{3!}}{\displaystyle \sum _{j_1=1}^{TP}{\displaystyle \sum _{j_2=1}^{TP}{\displaystyle \sum _{j_3=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^3{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_{j_1}\mathsf{\partial }{\alpha }_{j_2}\mathsf{\partial }{\alpha }_{j_3}}\right\}}_{{\alpha }^0}}}}\delta {\alpha }_{j_1}\delta {\alpha }_{j_2}\delta {\alpha }_{j_3}\\ +{\displaystyle \frac{1}{4!}}{\displaystyle \sum _{j_1=1}^{TP}{\displaystyle \sum _{j_2=1}^{TP}{\displaystyle \sum _{j_3=1}^{TP}{\displaystyle \sum _{j_4=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^4{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_{j_1}\mathsf{\partial }{\alpha }_{j_2}\mathsf{\partial }{\alpha }_{j_3}\mathsf{\partial }{\alpha }_{j_4}}\right\}}_{{\alpha }^0}}}}}\delta {\alpha }_{j_1}\delta {\alpha }_{j_2}\delta {\alpha }_{j_3}\delta {\alpha }_{j_4}+{\epsilon }_k.\end{array}$$

(1)

In Equation (1), the notation ${\left\{ \right\}}_{{\alpha }^0}$ indicates that the functional derivatives within the braces are computed at the known expected/nominal parameter values, which are denoted as ${\alpha }_i^0$ (using the superscript “0”); the corresponding column vector of nominal parameter values is denoted as ${\alpha }^0\triangleq {\left({\alpha }_1^0,\dots ,{\alpha }_{TP}^0\right)}^{†}$. The quantity ${\epsilon }_k$ comprises all quantifiable errors in the representation of the computed response $r_k\left(\alpha \right)$ as a function of the model parameters $\alpha \triangleq {\left({\alpha }_1,\dots ,{\alpha }_{TP}\right)}^{†}$. Vectors and matrices will be denoted using bold letters, while the dagger “$†$” will be used to denote “transposition”. The symbol “$\triangleq$” will be used to denote “is defined as” or “is by definition equal to”. The radius/domain of convergence of the series in Equation (1) determines the largest values of the parameter variations $\delta {\alpha }_j$ that are admissible before the respective series becomes divergent. In turn, these maximum admissible parameter variations limit the largest parameter covariances/standard deviations, which are acceptable for using the Taylor expansion for computing the moments of the distribution of computed responses.

2.1. Input to the 4th-BERRU-PM Methodology: 4th-Order Sensitivity and Uncertainty Analysis of Model Responses to Model Parameters

The moments of the computed model responses are obtained by using the Taylor-series expansion shown in Equation (1) of a response in terms of parameter variations. The expression of these computed response moments up to sixth-order moments and the parameter distribution and sensitivities can be found in Ref. [12]. The 4th-BERRU-PM methodology incorporates computed response moments up to fourth-order, as provided below.

(i)

The expected value, denoted as $E_c\left(r_k\right)$, of a computed response $r_k\left(\alpha \right)$, for $k=1,\dots ,TR$; the vector $E_c\left(r\right)$ of the computed responses is defined as follows: $E_c\left(r\right)\triangleq {\left[E_c\left(r_1\right),\dots ,E_c\left(r_k\right),\dots ,E_c\left(r_{TR}\right)\right]}^{†}$. Up to, and including, the fourth-order response sensitivities to parameters, the expected value of a computed response has the following expression obtained by formally integrating Equation (1) over the unknown distribution of parameters:

$$\begin{array}{l}{E}_c\left(r_k\right)=r_k\left({\alpha }^0\right)+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^{TP}{\displaystyle \sum _{j=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^2{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i\mathsf{\partial }{\alpha }_j}\right\}}_{{\alpha }^0}{c}_{ij}^{\alpha }}}+{\displaystyle \frac{1}{6}}{\displaystyle \sum _{i=1}^{TP}{\displaystyle \sum _{j=1}^{TP}{\displaystyle \sum _{\mathcal{l}=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^3{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i\mathsf{\partial }{\alpha }_j\mathsf{\partial }{\alpha }_{\mathcal{l}}}\right\}}_{{\alpha }^0}{t}_{ij\mathcal{l}}^{\alpha }}}}\\ \hspace{0.17em}+{\displaystyle \frac{1}{4!}}{\displaystyle \sum _{i=1}^{TP}{\displaystyle \sum _{j=1}^{TP}{\displaystyle \sum _{\mathcal{l}=1}^{TP}{\displaystyle \sum _{m=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^4{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i\mathsf{\partial }{\alpha }_j\mathsf{\partial }{\alpha }_{\mathcal{l}}\mathsf{\partial }{\alpha }_m}\right\}}_{{\alpha }^0}{q}_{ij\mathcal{l}m}^{\alpha }}}}}+\dots \end{array}$$

(2)

In Equation (2), the moments up to and including the fourth-order of the unknown distribution of model parameters are assumed to be known. These moments are as follows: (a) the covariances of two model parameters, ${\alpha }_i$ and ${\alpha }_j$, are denoted as $c_{ij}^{\alpha }$, $i,j=1,\dots ,TP$, where $TP$ denotes the total number of parameters under consideration; the parameter covariance matrix is denoted as $C_{\alpha \alpha }\triangleq {\left[c_{ij}^{\alpha }\right]}_{TP\times TP}$; (b) the triple-correlations of three model parameters ${\alpha }_i$, ${\alpha }_j$, and ${\alpha }_l$, are denoted as $t_{ij\mathcal{l}}^{\alpha }$, where $i,j,\mathcal{l}=1,\dots ,TP$; (c) the quadruple-correlations of four model parameters ${\alpha }_i$, ${\alpha }_j$, ${\alpha }_{\mathcal{l}}$, and ${\alpha }_m$, are denoted as $q_{ij\mathcal{l}m}^{\alpha }$, where $i,j,\mathcal{l},m=1,\dots ,TP$.

(ii)

The correlation, denoted as $\mathrm{cor}\left({\alpha }_i,r_k\right)$, between a parameter ${\alpha }_i$ and a computed response $r_k$, for $i=1,\dots ,TP$ and $k=1,\dots ,TR$; the correlation matrix between parameters and computed responses is denoted as $C_{\alpha r}^c\triangleq {\left[\mathrm{cor}\left({\alpha }_i,r_k\right)\right]}_{TP\times TR}$. Up to, and including, the fourth-order response sensitivities to parameters, the correlation between a parameter and a computed response has the following expression:

$$\begin{array}{l}\mathrm{cor}\left({\alpha }_i,r_k\right)={\displaystyle \sum _{j=1}^{TP}{\left\{\frac{\mathsf{\partial }{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_j}\right\}}_{{\alpha }^0}{c}_{ij}^{\alpha }}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^{TP}{\displaystyle \sum _{\mathcal{l}=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^2{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_j\mathsf{\partial }{\alpha }_{\mathcal{l}}}\right\}}_{{\alpha }^0}{t}_{ij\mathcal{l}}^{\alpha }}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \frac{1}{6}}{\displaystyle \sum _{j=1}^{TP}{\displaystyle \sum _{\mathcal{l}=1}^{TP}{\displaystyle \sum _{m=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^3{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_j\mathsf{\partial }{\alpha }_{\mathcal{l}}\mathsf{\partial }{\alpha }_m}\right\}}_{{\alpha }^0}}}}{q}_{ij\mathcal{l}m}^{\alpha }+\dots \end{array}$$

(3)

(iii)

The covariances, denoted as $\mathrm{cov}\left(r_k,\hspace{0.17em}{r}_{\mathcal{l}}\right)$, between two computed responses $r_k$ and $r_{\mathcal{l}}$, for $k,\mathcal{l}=1,\dots ,TR$; the covariance matrix of computed responses is denoted as $C_{rr}^c\triangleq {\left[\mathrm{cov}\left(r_k,\hspace{0.17em}{r}_{\mathcal{l}}\right)\right]}_{TR\times TR}$. Up to and including the fourth-order response sensitivities to parameters, the covariance between two computed responses has the following expression:

$$\begin{array}{l}\mathrm{cov}\left(r_k,\hspace{0.17em}{r}_{\mathcal{l}}\right)\hspace{0.17em}={\displaystyle \sum _{i=1}^{TP}{\displaystyle \sum _{j=1}^{TP}{\left\{\frac{\mathsf{\partial }{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i}\frac{\mathsf{\partial }{r}_{\mathcal{l}}\left(\alpha \right)}{\mathsf{\partial }{\alpha }_j}\right\}}_{{\alpha }^0}}{c}_{ij}^{\alpha }}\\ +{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^{TP}{\displaystyle \sum _{j=1}^{TP}{\displaystyle \sum _{m=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^2{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i\mathsf{\partial }{\alpha }_j}\frac{\mathsf{\partial }{r}_{\mathcal{l}}\left(\alpha \right)}{\mathsf{\partial }{\alpha }_m}+\frac{\mathsf{\partial }{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i}\frac{{\mathsf{\partial }}^2{r}_{\mathcal{l}}\left(\alpha \right)}{\mathsf{\partial }{\alpha }_j\mathsf{\partial }{\alpha }_m}\right\}}_{{\alpha }^0}}}}\hspace{0.33em}{t}_{ijm}^{\alpha }\\ +{\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^{TP}{\displaystyle \sum _{j=1}^{TP}{\displaystyle \sum _{m=1}^{TP}{\displaystyle \sum _{n=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^2{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i\mathsf{\partial }{\alpha }_j}\frac{{\mathsf{\partial }}^2{r}_{\mathcal{l}}\left(\alpha \right)}{\mathsf{\partial }{\alpha }_m\mathsf{\partial }{\alpha }_n}\right\}}_{{\alpha }^0}}}}}\left(q_{ijmn}^{\alpha }-c_{ij}^{\alpha }{c}_{mn}^{\alpha }\right)\\ +{\displaystyle \frac{1}{6}}{\displaystyle \sum _{i=1}^{TP}{\displaystyle \sum _{j=1}^{TP}{\displaystyle \sum _{m=1}^{TP}{\displaystyle \sum _{n=1}^{TP}{\left\{\frac{{\mathsf{\partial }}^3{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i\mathsf{\partial }{\alpha }_j\mathsf{\partial }{\alpha }_m}\frac{\mathsf{\partial }{r}_{\mathcal{l}}\left(\alpha \right)}{\mathsf{\partial }{\alpha }_n}+\frac{\mathsf{\partial }{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i}\frac{{\mathsf{\partial }}^3{r}_{\mathcal{l}}\left(\alpha \right)}{\mathsf{\partial }{\alpha }_j\mathsf{\partial }{\alpha }_m\mathsf{\partial }{\alpha }_n}\right\}}_{{\alpha }^0}}}}}\hspace{0.33em}{q}_{ijmn}^{\alpha }+\dots \end{array}$$

(4)

(iv)

The triple correlations among three responses, $r_k$, $r_{\mathcal{l}}$ and $r_m$, $k,\mathcal{l},m=1,\dots ,TR$, which are denoted as ${\mu }_3\left(r_k,r_{\mathcal{l}},r_m\right)$. Up to and including the fourth-order response sensitivities to parameters, these triple correlations among three responses have the following expression:

$$\begin{array}{l}{\mu }_3\left(r_k,r_{\mathcal{l}},r_m\right)={\displaystyle \sum _{i=1}^{TP}{\displaystyle \sum _{j=1}^{TP}{\displaystyle \sum _{\mu =1}^{TP}{\left\{\frac{\mathsf{\partial }{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i}\frac{\mathsf{\partial }{r}_{\mathcal{l}}\left(\alpha \right)}{\mathsf{\partial }{\alpha }_j}\frac{\mathsf{\partial }{r}_m\left(\alpha \right)}{\mathsf{\partial }{\alpha }_{\mu }}\right\}}_{{\alpha }^0}}{t}_{ij\mu }^{\alpha }}}\\ +{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^{TP}{\displaystyle \sum _{j=1}^{TP}{\displaystyle \sum _{\mu =1}^{TP}{\displaystyle \sum _{\nu =1}^{TP}\left\{\frac{\mathsf{\partial }{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i}\frac{\mathsf{\partial }{r}_{\mathcal{l}}\left(\alpha \right)}{\mathsf{\partial }{\alpha }_j}\frac{{\mathsf{\partial }}^2{r}_m\left(\alpha \right)}{\mathsf{\partial }{\alpha }_{\mu }\mathsf{\partial }{\alpha }_{\nu }}\right.\left(q_{ij\mu \nu }^{\alpha }-c_{ij}^{\alpha }{c}_{\mu \nu }^{\alpha }\right)}}}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\left.{\displaystyle \frac{\mathsf{\partial }{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i}}{\displaystyle \frac{{\mathsf{\partial }}^2{r}_{\mathcal{l}}\left(\alpha \right)}{\mathsf{\partial }{\alpha }_j\mathsf{\partial }{\alpha }_{\mu }}}{\displaystyle \frac{\mathsf{\partial }{r}_m\left(\alpha \right)}{\mathsf{\partial }{\alpha }_{\nu }}}\left(q_{ij\mu \nu }^{\alpha }-c_{i\nu }^{\alpha }{c}_{j\mu }^{\alpha }\right)\right\}}_{{\alpha }^0}+\dots \end{array}$$

(5)

(v)

The quadruple-correlations among four responses, $r_k$, $r_{\mathcal{l}}$, $r_m$ and $r_n$, for $k,\mathcal{l},m,n=1,\dots ,TR$, which are denoted as ${\mu }_4\left(r_k,r_{\mathcal{l}},r_m,r_n\right)$. Up to and including the fourth-order response sensitivities to parameters, these quadruple correlations among computed responses have the following expression:

$${\mu }_4\left(r_k,r_{\mathcal{l}},r_m,r_n\right)={\displaystyle \sum _{i=1}^{TP}{\displaystyle \sum _{j=1}^{TP}{\displaystyle \sum _{\mu =1}^{TP}{\displaystyle \sum _{\nu =1}^{TP}{\left\{\frac{\mathsf{\partial }{r}_k\left(\alpha \right)}{\mathsf{\partial }{\alpha }_i}\frac{\mathsf{\partial }{r}_l\left(\alpha \right)}{\mathsf{\partial }{\alpha }_j}\frac{\mathsf{\partial }{r}_m\left(\alpha \right)}{\mathsf{\partial }{\alpha }_{\mu }}\frac{\mathsf{\partial }{r}_n\left(\alpha \right)}{\mathsf{\partial }{\alpha }_{\nu }}\right\}}_{{\alpha }^0}}}}}{q}_{ij\mu \nu }^{\alpha }+\dots$$

(6)

(vi)

The expressions of the triple and quadruple correlations among parameters and responses are provided in Ref. [12]; they will not be reproduced here because they are considered to be negligible by comparison to the other terms used within the 4th-BERRU-PM methodology.

2.2. Applicability of the 4th-Order Sensitivity and Uncertainty Analysis to Energy Systems

The expressions of the moments of the distribution of computed responses provided in Equations (2)–(6) involve combinations of sensitivities of responses with respect to model parameters and moments of the distribution of parameters. Consequently, the computation of these moments is tantamount to performing both a “4th-order sensitivity analysis” (since one needs to compute the respective sensitivities) as well as a “4th-order uncertainty analysis” (since one determines the respective moments of the distribution of the computed response in the phase space of the model’s parameters) of the computational model under consideration. In principle, sensitivity and uncertainty analyses can be performed by using either deterministic or statistical methods. The statistical methods construct an approximate response distribution (often called “response surface”) in the parameters’ space by performing many “forward” computations using the model with altered parameter values, and subsequently use scatter plots, regression, rank transformation, correlations, and/or so-called “partial correlation analysis” in order to identify approximate expectation values, variances, and covariances for the responses. These statistical quantities are subsequently used to construct quantities that play the role of (approximate) first-order response sensitivities. Thus, statistical methods commence with “uncertainty analysis” and subsequently attempt an approximate “sensitivity analysis” of the approximately computed model “response surface”. Statistical methods for uncertainty and sensitivity analysis are reviewed in Ref. [16]. Although statistical methods for uncertainty and sensitivity analysis are conceptually easy to implement, they are subject to the curse of dimensionality [13] and cannot compute any sensitivity exactly. Also, since the response sensitivities and parameter uncertainties are inseparably amalgamated within the results produced by statistical methods, improvements in parameter uncertainties cannot be directly propagated to improve response uncertainties; rather, the entire set of simulations and statistical post-processing must be repeated anew. On the other hand, the computation by conventional deterministic methods of the n^th-order sensitivities (i.e., functional derivatives of a response with respect to the $TP$-parameters on which it depends) would also require at least $O\left(T{P}^n\right)$ large-scale computations, so these methods also suffer from the curse of dimensionality in sensitivity analysis.

Currently, the only methodologies that enable the exact and efficient computation of arbitrarily high-order sensitivities while overcoming the curse of dimensionality are the “n^th-order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems” (n^th-CASAM-L; Ref. [12]) and the “n^th-order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n^th-CASAM-N; Ref. [15]). These methods are applicable to compute exactly and efficiently arbitrarily-high order sensitivities of responses with respect to parameters for mathematical/computational models of any energy system. The higher the order of computed sensitivities, the higher the efficiency of the n^th-CASAM-L or n^th-CASAM-N methodologies by comparison to any other method. For example, as has been illustrated in Ref. [12] and as will be discussed in Section 3, below, the most important sensitivities of the PERP leakage response are with respect to the 180 uncertain total microscopic cross sections. For these sensitivities, the n^th-CASAM-L needs a single “large-scale” adjoint computation to obtain all of the 180 first-order sensitivities exactly, in comparison to needing 360 large-scale computations to obtain them inexactly, using finite differences. Similarly, the n^th-CASAM-L needs 2,075,341 large-scale computations to obtain the 45,212,895 distinct 4th-order sensitivities exactly, while using finite differences would require 723,404,160 large-scale computations to obtain them approximately.

3. Illustrative High-Order Uncertainty Analysis of the PERP Reactor Physics Benchmark

The Polyethylene-Reflected Plutonium (acronym: PERP) reactor physics benchmark [2] is a one-dimensional spherical subcritical nuclear system driven by a source of spontaneous fission neutrons. The computational model of the polyethylene PERP benchmark used in this work is the same as that presented in Ref. [12]. For convenience, the main features of this model are summarized in Appendix A. As discussed in Appendix A, this benchmark comprises 21,976 imprecisely known model parameters. The result (“response”) of interest for this benchmark is the total leakage of neutrons through the benchmark’s outer surface. Since the correlations between these parameters are unavailable, they will be considered to be uncorrelated and normally distributed, in accordance with the principle of Maximum Entropy of considering the least biased distribution based on the available information. Since the parameters are considered to be uncorrelated, the off-diagonal terms of $C_{\alpha \alpha }$ vanish, i.e., $c_{ij}^{\alpha }=0$ when $i\ne j$, while the diagonal terms are the variances of the respective parameters, denoted as $c_i^{\alpha }$, for $i=1,\dots ,TP$. Since the parameters are considered to be normally distributed, their third-order (triple) correlations vanish, i.e., $t_{ijk}^{\alpha }\triangleq cor\left({\alpha }_i,{\alpha }_j,{\alpha }_k\right)=0$ for all $i,j,k=1,\dots ,TP$. Furthermore, the only nonzero fourth-order (quadruple) correlation is the kurtosis of each individual parameter, which will be denoted as $q_i^{\alpha }$, $i=1,\dots ,TP$, which is related to the variance of the respective uncorrelated parameter as follows: $q_i^{\alpha }=3c_i^{\alpha }$, where $c_i^{\alpha }$ denotes the variance (i.e., standard deviation squared) of the parameter ${\alpha }_i$; all other quadruple correlations vanish, i.e., $q_{ijk\mathcal{l}}^{\alpha }=0$ if $i\ne j\ne k\ne \mathcal{l}$.

The comprehensive computation of response sensitivities with respect to the model parameters, up to and including fourth-order sensitivities, was reported in [12], where it was shown that the most important parameters are the 180 group-averaged microscopic total cross sections. As has been discussed in the forgoing, these parameters are considered to be uncorrelated; therefore, only the unmixed sensitivities are influential. The largest unmixed sensitivities occur for isotope 6 (¹H).

Table 1. Comparison of the unmixed relative sensitivities, $S^{(1)}\left({\sigma }_{t,6}^g\right)$, $S^{(2)}\left({\sigma }_{t,6}^g,{\sigma }_{t,6}^g\right)$, $S^{(3)}\left({\sigma }_{t,6}^g,{\sigma }_{t,6}^g,{\sigma }_{t,6}^g\right)$, and $S^{(4)}\left({\sigma }_{t,6}^g,{\sigma }_{t,6}^g,{\sigma }_{t,6}^g,{\sigma }_{t,6}^g\right)$, $g=1,\dots ,30$, for isotope 6 (¹H).

g	1st-Order	2nd-Order	3rd-Order	4th-Order
1	−8.471 × 10−6	7.636 × 10−7	6.322 × 10−8	1.460 × 10−7
2	−2.060 × 10−5	2.280 × 10−6	4.516 × 10−8	4.956 × 10−7
3	−6.810 × 10−5	9.021 × 10−6	−4.677 × 10−7	2.245 × 10−6
4	−3.932 × 10−4	6.673 × 10−5	−8.758 × 10−6	2.039 × 10−5
5	−2.449 × 10−3	5.549 × 10−4	−1.216 × 10−4	2.142 × 10−4
6	−9.342 × 10−3	2.935 × 10−3	−1.123 × 10−3	1.553 × 10−3
7	−7.589 × 10−2	3.949 × 10−2	−2.690 × 10−2	3.513 × 10−2
8	−9.115 × 10−2	5.604 × 10−2	−4.380 × 10−2	5.536 × 10−2
9	−1.358 × 10−1	1.014 × 10−1	−9.758 × 10−2	1.416 × 10−1
10	−1.659 × 10−1	1.428 × 10−1	−1.604 × 10−1	2.582 × 10−1
11	−1.899 × 10−1	1.849 × 10−1	−2.385 × 10−1	4.233 × 10−1
12	−4.446 × 10−1	6.620 × 10−1	−1.373 × 100	3.815 × 100
13	−5.266 × 10−1	9.782 × 10−1	−2.590 × 100	9.015 × 100
14	−5.772 × 10−1	1.262 × 100	−3.991 × 100	1.650 × 101
15	−5.820 × 10−1	1.391 × 100	−4.581 × 100	2.208 × 101
16	−1.164 × 100	4.460 × 100	−2.530 × 101	1.890 × 102
17	−1.173 × 100	4.853 × 100	−2.991 × 101	2.432 × 102
18	−1.141 × 100	4.828 × 100	−3.049 × 101	2.543 × 102
19	−1.094 × 100	4.619 × 100	−2.913 × 101	2.428 × 102
20	−1.033 × 100	4.284 × 100	−2.655 × 101	2.175 × 102
21	−9.692 × 10−1	3.937 × 100	−2.388 × 101	1.915 × 102
22	−8.917 × 10−1	3.515 × 100	−2.069 × 101	1.609 × 102
23	−8.262 × 10−1	3.177 × 100	−1.823 × 101	1.382 × 102
24	−7.495 × 10−1	2.792 × 100	−1.552 × 101	1.140 × 102
25	−7.087 × 10−1	2.604 × 100	−1.427 × 101	1.033 × 102
26	−6.529 × 10−1	2.349 × 100	−1.260 × 101	8.932 × 101
27	−5.845 × 10−1	2.039 × 100	−1.061 × 101	7.288 × 101
28	−5.474 × 10−1	1.885 × 100	−9.678 × 100	6.565 × 101
29	−5.439 × 10−1	1.891 × 100	−9.800 × 100	6.705 × 101
30	−9.366 × 100	4.296 × 102	−2.966 × 104	2.720 × 106

, below, presents a comparison of the values of the unmixed relative sensitivities, from first-order through fourth-order, for isotope 6 (¹H). Sensitivities that have absolute values larger than unity are presented in bold characters. As shown in Table 1, the largest absolute values for the 1st-, 2nd-, 3rd- and 4th-order unmixed relative sensitivities all occur for the lowest-energy group (g = 30; thermal neutrons), which are significantly larger than the values of the sensitivities in other energy groups. Notably, the largest 4th-order unmixed relative sensitivity attains a very large value: $S^{(4)}\left({\sigma }_{t,6}^{g=30},{\sigma }_{t,6}^{g=30},{\sigma }_{t,6}^{g=30},{\sigma }_{t,6}^{g=30}\right)=2.720\times {10}^6$. By comparison, the largest values for the 1st-, 2nd- and 3rd-order unmixed relative sensitivities are: $S^{(1)}\left({\sigma }_{t,6}^{g=30}\right)=-9.366$, $S^{(2)}\left({\sigma }_{t,6}^{g=30},{\sigma }_{t,6}^{g=30}\right)=4.296\times {10}^2$ and $S^{(3)}\left({\sigma }_{t,6}^{g=30},{\sigma }_{t,6}^{g=30},{\sigma }_{t,6}^{g=30}\right)=-2.966\times {10}^4$, respectively.

The effects of various values for the standard deviations for the parameters, which are considered to be uncorrelated and normally distributed, will be illustrated in Section 3.1, Section 3.2 and Section 3.3, below, for parameters that are known with “high precision” (having uniform standard deviations of 2%); parameters known with “medium precision” (having uniform standard deviations of 5%); and parameters known with “low precision” (having uniform standard deviations of 10%), respectively.

3.1. “High Precision” Parameters, Having Uniform Relative Standard Deviations

As has been discussed in [12], when the uncorrelated and normally-distributed parameters are assumed to have uniform relative standard deviations of 3%, the convergence “ratio-test” of the 3rd-order term with respect to the 2nd-order term of the Taylor series is 0.58, while the ratio of the 4th-order term with respect to the 3rd-order term of the Taylor series is 0.68. Both of these results are below 1.00, which implies that for uniform relative standard deviations of 3% or less, the Taylor-series expansion of the computed response in terms of the model parameters shown in Equation (1) is expected to be convergent. Figure 1 depicts the magnitudes of the higher-order contributions to the expected value, $E(L^c)$, of the computed leakage response, while Figure 2 depicts the magnitudes of the higher-order contributions to the variance, $\mathrm{var}(L^c)$, of the computed leakage response, for parameters that are assumed to have uniform relative standard deviations of 2% (which is smaller than 3%). The numerical results depicted in Figure 1 and Figure 2 indicate that the contributions of the increasingly higher-order terms to the expected value, $E(L^c)$, and the variance, $\mathrm{var}(L^c)$, of the computed leakage response become increasingly smaller (as the order of the respective terms increases), thus confirming the expectation that the underlying Taylor-series is convergent. For practical purposes, therefore, the contributions from terms involving sensitivities of order five and higher become negligible by comparison to the contributions from the terms comprising the sensitivities of first-through fourth-order to $E(L^c)$ and $\mathrm{var}(L^c)$, respectively, when the uncorrelated and normally-distributed parameters have uniform relative standard deviations $SD=2\%$.

3.2. “Medium Precision” Parameters, Having Uniform Relative Standard Deviations

For uniform relative standard deviations of 5% for the uncorrelated and normally-distributed model parameters, it has been shown in [12] that the ratio of the 3rd-order term with respect to the 2nd-order term of the Taylor series is 0.97 < 1.00, but the ratio of the 4th-order term with respect to the 3rd-order term of the Taylor series is 1.13 > 1.00. These ratios indicate that relative standard deviations of 5% for the model parameters are outside of the radius of convergence of the Taylor-series presented in Equation (1). These indications are confirmed by the results depicted in Figure 3 and Figure 4, below.

The results depicted in Figure 3 indicate that the contributions to $E(L^c)$ stemming from second-order sensitivities are smaller than those stemming from the zeroth-order term, $L^c$, but the contributions to $E(L^c)$ stemming from fourth-order sensitivities are larger than those stemming from the zeroth- and second-order terms. This oscillatory behavior with increasing amplitudes is indicative of the divergence of the Taylor-series underlying the computation of the expected value, $E(L^c)$. The results depicted in Figure 4 for the variance, $\mathrm{var}(L^c)$, of the computed leakage response indicate that the contributions to $\mathrm{var}(L^c)$ increase as the order of the contributing terms increase, thus underscoring the divergent nature of the underlying Taylor-series when the parameters have uniform relative standard deviations of 5%. On the other hand, relative standard deviation of 5% are often encountered in measurements of total cross sections, which highlights the need for computing the second and higher-order response sensitivities in order to investigate the convergence properties of the Taylor-series that underlies the determination of the statistics (expected values, variance, etc.) of the distribution of computed responses in the phase-space of imprecisely known model parameters.

3.3. “Low Precision” Parameters, Having Uniform Relative Standard Deviations

When considering uniform relative standard deviations of 10% for the uncorrelated and normally-distributed parameters (total cross sections), it has been shown in [12] that the ratio of the 3rd-order term with respect to the 2nd-order term of the Taylor series is 1.93; the ratio of the 4th-order term with respect to the 3rd-order term of the Taylor series is 2.26. Both of these results are larger than 1.00, indicating that the Taylor-series presented in Equation (1) would be divergent if used for parameters having standard deviations of 10%. The divergence of the Taylor-series for such parameter standard deviations is underscored by the corresponding results depicted in Figure 5 for the expected value, $E(L^c)$, of the computed leakage response, which clearly indicate the massive increase of the contributions to $E(L^c)$ as the order of the retained terms increases. The conclusion that the Taylor-series expansion is divergent and should therefore not be used for parameters with uniform relative standard deviations $SD=10\%$ is reinforced by the corresponding results depicted in Figure 6 for the variance, $\mathrm{var}(L^c)$, of the computed leakage response. Figure 6 also highlights that the contributions to $\mathrm{var}(L^c)$ increase massively as the order of the retained terms increases.

4. Concluding Remarks

This work has reviewed the fourth-order “sensitivity analysis” and “uncertainty quantification” aspects of computational models, the results of which are used as “input” into the 4th-BERRU-PM methodology. The impact of combinations of sensitivities of increasingly higher order and various values for the standard deviations of the model’s parameters has been illustrated by using the PERP reactor physics benchmark. This benchmark is modeled by the neutron transport equation comprising 21,976 model parameters and is therefore representative of “large-scale” computational models of energy systems. It has been shown that the series-expansion representation of the expected value and variance of the computed leakage response is convergent and hence produces reliable results for normally distributed parameters with uniform relative standard deviations of 2%. On the other hand, the series-expansion representations of the expected value and variance, respectively, become divergent for parameters having uniform relative standard deviations of 5%. This divergence becomes massive for parameters with uniform relative standard deviations of 10%.

In the accompanying Part 2 [14], the results obtained in this work will be combined, using the maximum entropy principle within the 4th-BERRU-PM methodology, with the first four moments of the distribution of measured responses to obtain the best-estimate predicted mean value, standard deviation, skewness, and kurtosis for the neutron leakage response of the PERP benchmark, thereby illustrating the applicability of the 4th-BERRU-PM methodology to improve the predictability and accuracy of models and data. The accuracy improvement in both the results and the data arises from the underlying MaxEnt-root of the 4th-BERRU-PM methodology, which enables this methodology to be used simultaneously for forward and inverse predictions. It will be specifically shown that the improvement of the data (i.e., parameters) used in a model comprises the best-estimate values predicted for the data along with reduced predicted standard deviations for this data, as provided by the 4th-BERRU-PM methodology in the “inverse predictive” mode. Simultaneously, improvements in the predictions of models are enabled by the use of the 4th-BERRU-PM methodology in the “forward predictive” mode, which will be shown to provide best-estimate model responses (results) along with reduced standard deviations for these predicted responses. Of course, the predicted responses will also be improved by using the models with the improved, calibrated data, obtained after having applied the 4th-BERRU-PM methodology.